Question

Which of the following option correctly describes the variation of the speed \(v\) and acceleration 'a' of a point mass falling vertically in a viscous medium that applies a force \(F=-kv\), where 'k' is a constant, on the body ? (Graphs are schematic and not drawn to scale)

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

Write \(a=g-\frac{k}{m}v\) and note that a decreases as v increases up to terminal velocity.

Answer 1

The Correct Option is B

Use the slope relation: \(a=\frac{dv}{dt}\). Since \(a=g-\frac kmv\), a is largest when \(v=0\) and falls to zero when v reaches \(\frac{mg}{k}\); the slope of the v-t curve keeps decreasing, so the v-t curve is concave and flattens.
Because \(\frac{da}{dt}=-\frac kma\), the rate of fall of a is proportional to a itself, an exponential decay (not a straight line).
Only graph (2) shows v increasing and saturating with a decaying exponentially.
Answer: (2) v rises from 0 to a terminal value and a decays exponentially from g to 0